# Probe 5: Noise vs Signal Decomposition

Comparing 69-country vs 140-country samples.
SE² = σ² × (X'X)⁻¹, so SE change decomposes into residual variance × leverage.

## Simple (4 ctrls)

| Metric | 69-country | 140-country | Δ | % change |
|--------|------------|-------------|---|----------|
| β(Z₁) | 5.6640 | 23.2139 | +17.5499 | +309.8% |
| SE(Z₁) | 16.6228 | 12.6219 | -4.0009 | -24.1% |
| p-value(Z₁) | 0.7333 | 0.0660 | -0.6674 | — |
| σ² (residual var) | 30.0414 | 80.5391 | +50.4977 | +168.1% |
| (X'X)⁻¹[Z₁,Z₁] | 1.7822 | 0.6391 | -1.1431 | -64.1% |
| N obs | 1979.0000 | 3869.0000 | +1890.0000 | +95.5% |
| R² | 0.2747 | 0.2244 | -0.0503 | — |

## Full (7 ctrls)

| Metric | 69-country | 140-country | Δ | % change |
|--------|------------|-------------|---|----------|
| β(Z₁) | 15.2579 | 19.0299 | +3.7721 | +24.7% |
| SE(Z₁) | 21.7433 | 16.1182 | -5.6251 | -25.9% |
| p-value(Z₁) | 0.4830 | 0.2378 | -0.2451 | — |
| σ² (residual var) | 29.3909 | 54.8620 | +25.4711 | +86.7% |
| (X'X)⁻¹[Z₁,Z₁] | 3.0765 | 1.0328 | -2.0438 | -66.4% |
| N obs | 1316.0000 | 2695.0000 | +1379.0000 | +104.8% |
| R² | 0.3952 | 0.2836 | -0.1117 | — |

